Juan Rivera-Letelier

Théorie ergodique des fractions rationnelles sur un corps ultramétrique

We make the first steps towards an understanding of the ergodic properties of a rational map defined over a complete algebraically closed non-archimedean field. For such a rational map R, we construct a natural invariant probability measure m_R which reprensents the asymptotic distribution of preimages of non-exceptional point. We show that this measure is exponentially mixing, and satisfies the central limit theorem. We prove some general bounds on the metric entropy of m_R, and on the topological entropy of R. We finally prove that rational maps with vanishing topological entropy have potential good reduction.

A connecting lemma for rational maps satisfying a no-growth condition

We introduce and study a non-uniform hyperbolicity condition for complex rational maps that does not involve a growth condition. We call this condition backward contraction. We show this condition is weaker than the Collet?Eckmann condition, and than the summability condition with exponent one. Our main result is a connecting lemma for backward-contracting rational maps, roughly saying that we can perturb a rational map to connect each critical orbit in the Julia set with an orbit that does not accumulate on critical points. The proof of this result is based on Thurstons algorithm and some rigidity properties of quasi-conformal maps. We also prove that the Lebesgue measure of the Julia set of a backward-contracting rational map is zero, when it is not the whole Riemann sphere. The basic tool of this article is sets having a Markov property for backward iterates that are holomorphic analogues of nice intervals in real one-dimensional dynamics.

Equality of pressures for rational functions

We prove that for all rational functions f on the Riemann sphere and potential −t ln |f � | ,t ≥ 0 all the notions of pressure introduced in Przytycki (Proc. Amer. Math. Soc. 351(5) (1999), 2081-2099) coincide. In particular, we get a new simple proof of the equality between the hyperbolic Hausdorff dimension and the minimal exponent of conformal measure on a Julia set. We prove that these pressures are equal to the pressure definedwith theuse of periodicorbitsunderan assumptionthat therearenotmanyperiodic orbits with Lyapunov exponent close to 1 moving close together, in particular under the Topological Collet-Eckmann condition. In Appendix A, we discuss the case t< 0.

Weak hyperbolicity on periodic orbits for polynomials

Abstract We prove that if the multipliers of the repelling periodic orbits of a complex polynomial grow at least like n5+e with the period, for some e>0, then the Julia set of the polynomial is locally connected when it is connected. As a consequence for a polynomial the presence of a Cremer cycle implies the presence of a sequence of repelling periodic orbits with “small” multipliers. Somewhat surprisingly the proof is based on measure theorical considerations. To cite this article: J. Rivera-Letelier, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1113–1118.

Une caractérisation des fonctions holomorphes injectives en analyse ultramétrique

We prove that a non constant holomorphic function f defined over an analytic subspace of Cp is injective if and only if nf(x)−f(y)x−y2=f′(x)·f′(y),for every distinctxandy. nThis characterization proves the analogue, for holomorphic functions, of a conjecture of A. Escassut and M.C. Sarmant. On the other hand we give a counter-example to this conjecture, that concerns bi-analytic elements. To cite this article: J. Rivera-Letelier, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 441–446.